Reducing zero offset of MEMS sensors under thermal hysteresis conditions

Introduction

One of the problems faced by aerial vehicles is equipment tolerance to abrupt changes of temperature depending on the time of day and altitude. This problem is especially characteristic of MEMS sensors employed in navigation systems. It is known that the calibration algorithms and data suitable for mildly varying external effects are not acceptable for the conditions of extreme change of temperatures [1]. However, such process can be simulated in laboratory conditions and taken into account during calibration [2].

There is a number of papers focusing on the dependence of MEMS sensors’ errors, such as zero drift and scale factor instability, on temperature. Thus, paper [3] investigated into variation in the position of silicon structure points of a sensing element; in the process, a difference of the order of 100 nm was discovered in this variation in the course of heating and cooling (Fig. 1).

It testifies to different thermal deformation of silicon components, depending on different thermal dynamics conditions. Variation in the dimensions of the sensing masses leads to a change of the natural frequencies of their oscillations and, as a consequence, to somewhat changing accuracy characteristics. It should also be mentioned that MEMS capsule heats up and cools down non-uniformly. Given as an example in Fig. 2 is cooling of an MEMS capsule in its installation point [4].

This introduces additional dependence on the thermal dynamics.

However, said peculiarities can be taken into account even when working with MEMS sensor as the end product, i.e. not interfering with its inner design and adjustment. Thus, paper [5] considered dynamic correction of drift at different temperatures caused by inner heating of electronic components. However, this paper, while distinguishing between the initial offset and drift (change of displacement in the course of sensor operation), focuses on the dynamics of the second parameter only and does not consider the hysteresis effect, i.e. change in external impacts, which is undetermined at the outset. Paper [2] accounts for sensor internal heating, drift variance vs. time under oppositely directed hysteresis, and also temperature variation magnitude. Paper [6] offers a linear interpolation of hysteresis drift, and in [7, 8] a neural network mechanism for estimating drift behaviour during thermal cycling is proposed. Still, no paper provides a universal algorithm accounting for the dynamics under all possible thermal conditions.

Problem statement

On the whole, in terms of statistical diversity and dependence on conditions, zero offset can be resolved into components according to the following model (the example is given for gyroscopes; for accelerometers, the model is similar):

Δω_см = Δω_Т.сист + Δω_хр + Δω_t+ Δω_нестаб, (1)

where Δω_Т.сист – zero offset systematic component, temperature-dependent;
Δω_хр – zero offset systematic component varying with the retention time;
Δω_t – zero offset systematic component dependent on the time from the moment of activation, under unchanged external conditions;
Δω_нестаб – zero offset instability from activation to activation.

The objective of this paper is a study of component Δω_Т.сист and specific features of its variability and application. At the moment of calibration we proceed from the assumption that Δω_хр = 0, and Δω_нестаб is a random process with zero mean value at a sufficient number of measurements.

Given the above specifics of MEMS sensors offset dependence not only on the current temperature of component elements, but also on the history of temperature change, it is necessary to propose a technological approach for determining the parameters of zero offset component Δω_Т.сист under different thermal dynamics conditions, identifying those conditions, and offering techniques for accounting for those parameters during calibration of sensors in a gyro inertial unit.

Investigated GIU and its properties

Calibration of MEMS sensors is performed on the sensors installed in a gyro inertial unit (GIU, Fig. 3), which can output both direct measurements of gyroscopes and accelerometers, and the navigation parameters, such as spatial orientation angles, linear velocities, and cardinal coordinates. The GIU contains a micro controller, which ensures measurement information pickup from the sensors, application of calibration algorithms and data, and output of the final parameters in a required form. The operating range of gyroscopes is ±500°/s, accelerometers – ±100 g, information output frequency – 1000 Hz.

Main points

Besides the disaggregation as per formula 1, in zero offset it is possible to distinguish the initial offset and drift of zero; such disaggregation of zero offset components is conditioned by physical reasons. The main cause of the initial offset variance is the instability of electronic signal converters and the error of ADC included in the MEMS sensor set, and that of the drift variance – in the instability of temperature gradients [9] inside the MEMS capsule. Both parameters can be described independently using formula 1 (for the initial offset, without consideration of Δω_t = 0), with somewhat different characteristics at that. Thus, both parameters have instability associated with longterm retention [10], with instability of the initial zero offset being considerably higher. Fig. 4а shows scattering of the initial offsets of MEMS gyroscopes, and Fig. 4b – similar scattering of the drifts of MEMS gyros (the drift is reckoned taking into account the deducted initial zero offset). It can be seen that scattering of the initial offset for gyroscopes is basically the same as scattering of the drifts, with account for the deducted initial offset. That is, the offset during the first two seconds has the same share in the total error as the offset during he next 40–60 seconds without consideration of the first two. A pattern like this holds true for the majority of MEMS sensors from different manufacturers, both domestic and foreign ones.

For constant temperatures, the problem of obtaining systematic calibration coefficients is solved by multiply repeating drift measurement under similar independent conditions, followed by averaging of the values. When measuring zero drift under hysteresis conditions, it is impossible to reach such conditions due to non-uniformity of the thermal dynamics provided by heat chamber, as well as mismatch between the temperature set by heat chamber and the actual temperature on the sensor (the reason for that is inertia of air temperature near the GIU sensors and internal heating of sensors during operation).

For solving the problem of hysteresisdependent zero offset, a methodology was developed, discriminating natural heating and drift induced by it from external heating and hysteresis drift, as well as accounting for different initial zero offsets. Time intervals corresponding to each temperature point within a range of 20...60 degrees with a 2-degree increment during temperature rise and drop were approximately determined through experiments. In the recurrent experiments, temperature values measured with thermal sensor in one point have RMSD within approximately 0.5 °C. Scattering of the offsets and drifts under conditions of a varying environment was somewhat higher than under steady external conditions.

The hysteresis effect manifests itself under abrupt (at least 2 degrees per minute) change of external (relative to sensor) environment. The temperature actually measured by the thermal sensor, when activated at cooling and heating, will be the same, just like in case of steady-state environment, but the thermal dynamics in the course of measurement will be different. Fig. 5 shows the dynamics of zero offset change under abrupt heating and cooling.

It can be seen from the figure that it is not only the drift that changes from the thermal dynamics, but also the initial offset, corresponding to which is the left end of the red line and the right end of the blue line in the two figures (after reaching +65 °С, power was supplied again). It means that the initial offset cannot be predicted right at the power-on, and some minimally reliable information on the difference between the temperature during measurement and the power-on temperature is required (it was established experimentally that the necessary time is at least 5 seconds). This is a sufficient time for reliable assessment of variations in the thermal sensor readings, by which it will be possible to identify the character of external temperature and select a corresponding operation mode: steady-state conditions, heating, or cooling. Only following this the coefficients adequate to the current dynamics can be elaborated on; at the same time, over a period of the first 5 seconds or more, an error may accumulate, critical to the navigation tasks, especially taking into account the high weight of the initial offset in the total error.

For GIUs with outputs carrying direct measurement information from gyroscopes and accelerometers, the offset can be estimated when taking measurements in the static position, using the formulas with linear time dependence:

Δω_{Т.систГ} = drg × t + ω₀, (2)
Δω_{Т.систА} = drg × t + n₀, (3)

where ω₀ and n₀ – initial zero offset of gyroscope and accelerometer, respectively;
dr – time-dependent drift change coefficient (for a case with several time points, a corresponding coefficient is calculated for each interval),
t – time from power-on.

In other words, formulas 2 and 3 offer determination of coefficients Δω_Т.сист separately, both for the initial offset and zero drift. In so doing, it is necessary to measure and take into account component Δω_t beforehand, which will ensure observability of Δω_Т.сист.

For GIUs with outputs carrying navigation information, offset in the static position is determined by a simplified formula [11], formula 4 serving as an example for the northern channel:

 (4)

where x₁ – coordinate error, x₂ – linear velocity error, x₃ – orientation angle error, x₄ = υω_Г – gyroscope zero offset, x₅ = υn_A – accelerometer zero offset, ω_ш – Schuler frequency, g – free fall acceleration, δω_Г – gyroscope noise, δn_A – accelerometer noise.

If the computing resources allow to determine calibration coefficients in the course of measurements rather than during post-processing, a Kalman filter can be applied [12]. An example of determining three drift coefficients in real time with the use of a Kalman filter is given in Fig. 6.

To account for scattering of the initial zero and drift offset values from activation to activation using the confidence interval formula, the number of recurrences was determined (15) for finding a systematic component of zero drift, with the confidence coefficient of 99 %. In case of 15 drift recurrences, the temperature values and correlation coefficients are linked by the following system of equations:

dr = T × drt, (5)

where drt – set of the coefficients of sensor drift vs. temperature;
T – set of the differences of temperatures measured by thermal sensor during measurement and at power-on.

The task is essentially an elementary linear regression whose sought parameter is determined by the least squares method [13]:

drt = (T^T × T) ^–1 × T^T × dr. (6)

A similar formula should be used for the initial zero offset as well. It is necessary to apply such an approach, which implies finding coefficients drt, for cases with constant temperature, heating, and cooling, because, proceeding from the data in Fig. 5, those coefficients will differ.

Application of this approach to offset compensation under thermal hysteresis conditions is complicated by the circumstance that it is a non-zero period of time elapsing before the moment when the coefficient can be accurately determined (as an example, we take a time point at the 10th second). Since the initial zero offset coefficient and the first drift coefficients, as determined for the current dynamics at the 10th second, may differ from the coefficients used before the 10th second due to preliminary estimation data, then, when using navigation parameters, a correction shall be introduced, accounting for the difference between the preliminary and final correlation coefficients. Depending on the available memory and computing resources, the following methods can be used.

1. If sufficient memory and computing resources are available, all the measured data can be saved in memory, and computations as per general formula of transition from the primary information sensor measurements to the navigation parameters can be performed anew. In that case, if the difference between the initial and the estimated parameter is considerable, an abrupt jump in the navigation parameter values may occur in the required accuracy achieving point as compared with the previous value. This may have negative consequences for the control system, therefore such method is inadvisable.
2. If no sufficient memory is available, method 1 can be simplified by substituting the measured data array for several averaged values. This yields a greater end error of the navigation parameters and does not solve the problem of parameters jump.
3. In the absence of substantial computing resources, a simplified coordinates error estimation formula for the inertial navigation systems can be used [14]:

where δx – coordinate error,
R – Earth’s radius,
φ – roll angle,
t – error estimation time,
∂V_x – derivative (change) of angular velocity eastern channel,
dr_ω – estimated drift value,
dr_ω0 – initial zero offset,
ω₀ – Schuler frequency,
dr_a – accelerometer (if available) drift,
β – pitch angle.

Such formula can be applied for each measurement and re-estimated drift value dr_ω, as well as re-estimated dr_ω0. Such formulas do not account for cross-couplings having an insignificant component, thus reducing computation from matrix equations to a linear formula, but they may have a considerable error over a long time interval.

Results of proposed algorithm application

The proposed method for compensation of zero offset in MEMS sensors was implemented algorithmically in the gyro inertial unit’s firmware. Application of this algorithm was evaluated after two weeks from calibration for a static position and output information with direct measurements of angular velocity and linear acceleration. The experiment was performed at reaching the temperature of +40 °C in three modes: constant temperature maintained, heating from +20 to +60 °C, and cooling from +60 to +20 °C (the instrument was activated at a temperature of +40 °C reached by the internal thermal sensor). This test was run 15 times, arithmetic mean values of gyroscope readings during 1 minute of operation were assessed, and the maximum value in modulo was taken as the final result. The results of the experiment are given in Table 1.

It can be seen from the results that even after calibration, the offset values under hysteresis conditions remained worse than at constant temperature. This is indicative of high variability of the parameters dependent on thermal dynamics. In other words, instability component Δω_нестаб from activation to activation is higher under the hysteresis conditions than at constant temperature. It can be noticed, too, that a certain constant component is present in all the results, which is explained by a change depending on retention time of Δω_хр. However, the value of this component was assessed for constant external conditions only, and its variation for the thermal hysteresis conditions requires additional research.

For checking the methods of applying the proposed algorithms to a unit with inertial navigation system outputs without correction, a semirealistic simulation was performed. The essence of simulation consisted in application of a model of sensors with zero offset parameters, which are close to the considered ones, as primary information sensors’ readings for an inertial navigation system modelled according to the traditional orientation and navigation algorithms. The study implied an option of navigation system static position; system state was assessed at the 10th second of its operation, for which a transition under the first and second method was algorithmically specified. The coordinate errors before and after transition, as well as a jump of parameters, are given in Table 2.

Evidently, method 3 has quite a considerable residual error, but features an advantage associated with the absence of jump. These results are indicative of additional complications faced when using navigation systems on MEMS sensors under abruptly changing external conditions. One of the solutions to this problem can be prediction of temperature variation depending on the motion conditions.

Conclusions

The paper describes and investigates the effect of hysteresis in zero offset of MEMS sensors. Zero offset disaggregation into components dependent on different conditions is proposed, and the essence of distinguishing between initial zero offset and zero drift is exemplified. The role of temperature component in zero offset total error is shown. A method for compensation of the initial zero offset and zero drift of MEMS sensors under the thermal hysteresis conditions is described, the method being a generalisation of the traditional method of zero offset calibration at constant external temperature. The novelty of the work is the proposed method of calibration, which enables to account for different zero offset in gyroscopes and accelerometers under initially undetermined thermal dynamics conditions. Three methods are described for eliminating the error of initially undetermined drift of MEMS sensors, when they are applied in an inertial navigation system. The proposed method for eliminating temperature component of zero offset of MEMS sensors can be useful when the latter are applied in the navigation system of highly dynamic aerial vehicles.